How can I animate a mesh moving in a spiral fashion?
Plane with Curve Modifier animated with Curve Location



The Curve is animated on the X axis with 2 location keyframes.



Spiral, Plane, Lattice in the image above.



The paper in edit mode, high density, loop cuts in both directions to suit your needs.  The Curve Modifier on the paper performs the rolling.



2 options at least.

The spiral moved on the x axis toward the paper.



The paper moved on the x axis toward the spiral.  



The spiral is the parent of the lattice which slightly squeezes the middle.  The lattice is optional and influence can be animated.  The spiral was created with the standard addon AddCurve extra objects.  The tail was extended on the x-axis to control the tangent and flatness in the XY plane. The origin was moved to the last point of curve.



In the picture above you see the origins of the spiral and the paper.  Be very aware of those locations.



More middle squeeze with Lattice.  The origin of the lattice is in the center of Spiral.

GIF created with GIMP.  Please excuse the jump.